《Finite element
methods（有限元方法）》系统地论述了有限元方法的数学基础理论。以椭圆偏微分方程边值问题为例，介绍了协调有限元方法以及非协调等非标准有限元方法的数学描述、收敛条件和性质、有限元解的先验和后验误差估计以及有限元空间的基本性质，其中包括作者多年来的部分研究成果。
目錄：
Preface to the Series in Information and Computational
Science
Preface
Chapter 1 Variational Principle
1.1 Sobolev Space
1.2 Poisson Equation
1.2.1 Dirichlet Problem
1.2.2 Neumann Problem
1.3 Biharmonic Equation
1.4 Abstract Variational Problem
1.5 Galerkin Method and Ritz Method
Chapter 2 Finite Element and Finite Element Space
2.1 Triangulation
2.2 Finite Element
2.3 Finite Element Space
2.4 Second Order Problem:Simplex Elements
2.4.1 Simplex Element of Degree k
2.4.2 Linear Simplex Element
2.4.3 Quadric Simplex Element
2.4.4 Cubic Simplex Element
2.4.5 Incomplete Cubic Simplex Element
2.4.6 Crouzeix-Raviart Element
2.4.7 Cubic Hermite Simplex Element
2.4.8 Zienkiewicz Element
2.5 Second Order Problem:Rectangle Elements
2.5.1 Rectangle Element of Typek
2.5.2 Incomplete Rectangle Element of Type2
2.5.3 Wilson Element
2.5.4 Rectangle C-R Element
2.6 Fourth Order Problem:Simplex Elements
2.6.1 Morley Element
2.6.2 Zienkiewicz Element
2.6.3 Morley-Zienkiewicz Element
2.6.4 Modified Zienkiewicz Element
2.6.5 12-parameter Triangle Plate Element
2.6.6 15-parameter Triangle Plate Element
2.6.7 Argyris Element
2.6.8 Bell Element
2.6.9 Cubic Tetrahedron Element
2.7 Fourth Order Problem:Rectangle Elements
2.7.1 Rectangle Morley Element
2.7.2 Adini Element
2.7.3 Bogner-Fox-Schmit Element
2.8 2m-th Order Problem:MWX Element
Chapter 3 Interpolation Theory of Finite Elements
3.1 Affne Mapping and Affne Family
3.2 Affne Continuity and Scale Invariance
3.3 Interpolation Error
3.4 Inverse Inequality
3.5 Approximate Error of Finite Element Spaces
3.6 Interpolation Error of General Element
Chapter 4 Conforming Finite Element Method
4.1 Poisson Equation
4.2 Plate Bending Problem
4.3 A Posteriori Error Estimate
Chapter 5 Nonconforming Finite Element Methods
5.1 Nonconforming Finite Element
5.2 Weak Continuity
5.3 Second Order Elliptic Problem
5.4 Fourth Order Elliptic Problem
5.5 2m-th Order Elliptic Problem
5.6 A Posteriori Error Estimate
5.7 Error Estimate in L2 Norm
Chapter 6 Convergence of Nonconforming Finite Element
6.1 Generalized Path Test
6.2 Patch Test
6.2.1 Patch Test
6.2.2 Weak Patch Test
6.2.3 Suffciency of Patch Test
6.2.4 Necessity of Patch Test
6.3 Counter Examples of Patch Test
6.4 F-E-M Test
6.4.1 F1-Test
6.4.2 F2-Test
6.4.3 E1-M1-Test
6.4.4 E2-M2-Test
6.4.5 Procedure of F-E-M Test
6.4.6 Quadrilateral Wilson element
6.4.7 8-parameter quadrilateral element
6.4.8 15-parameter triangle plate element
6.5 Superapproximation
6.6 Strange Convergence Behaviour
6.6.1 Second Carey Example
6.6.2 Zienkiewicz element
Chapter 7 Quasi-Conforming Element Method
7.1 Second Order Problem:RQC4 Element
7.2 Biharmonic Equation
7.2.1 TQC9 Element
7.2.2 TQC12 Element
7.2.3 TQC15 Element
7.2.4 RQC12 Element
7.2.5 Three Dimensional Case:TQC16 Element
7.3 Rank Condition
7.4 Approximation
7.5 Error Estimate
7.6 A Posteriori Error Estimate
Chapter 8 Unconventional Finite Element Method
8.1 Free Formulation Scheme
8.2 Two Finite Elements
8.2.1 TRUNC Element
8.2.2 Bergan Element
8.3 Convergence Analysis
8.4 General Situation
8.5 A Posteriori Error Estimate
Chapter 9 Double Set Parameter Method
9.1 DSP Method
9.2 Convergence of DSP Method
9.3 DSP Elements for Poisson Equation
9.4 DSP Elements for Plate Bending Problem
9.4.1 9-parameter Generalized Conforming Element
9.4.2 VZ1 Element
9.4.3 Variants of VZ1 Element
9.4.4 VZ2 Element
9.4.5 DSPT12 Element
9.4.6 Error Estimate
9.5 A Posteriori Error Estimate
Chapter 10 Property of Finite Element Space
10.1 Basic Assumptions
10.2 Embedding Property
10.3 Compact Property
10.4 Inequalities on Finite Element Spaces
10.5 Inequality About Maximum Norm
Chapter 11 L∞ Error Estimate for Second Order
Problem
11.1 Weighted Norm
11.2 Regular Green Function
11.3 Conforming Elements
11.4 Nonconforming Elements
Chapter 12 L∞ Error Estimate for Plate Bending
Problem
12.1 Regular Green Function
12.2 Conforming Element
12.3 Nonconforming Element
12.4 Quasi-Conforming Element
12.5 Unconventional Element
12.6 DSP Element
Bibliography
Index
內容試閱：
Chapter 1
Variational Principle
The.niteelementmethodisbaseonthevariationalformulaofpartialdi.erentialequation.Thepaper[1]ofFeng’sworkaboutthe.niteelementmethodwasjustentitledby‘Adi.erenceformulationbasedonthevariationalprinciple’.Forthevariationalproblemcorrespondingtothepartialdi.erentialequation,therearetwoclassicalapproximatemethods,theRitzmethodandtheGalerkinmethod.The.niteelementmethodisobtainedwhenthe.niteelementspacesareusedasthe.nitedimensionalspacesintheRitzmethodorintheGalerkinmethod.TheknowledgeabouttheSobolevspacesispreliminarytothemathematicaltheoryof.niteelementmethod.Section1.1willgivetheessentialresultsaboutSobolevspaceswithoutproof.Section1.2andSection1.3willintroducethevariationalformsofboundaryvalueproblemsofthePoissonequationandthebiharmonicequationrespectively.Section1.4willconsidertheLax-Milgramlemmaabouttheabstractvariationalproblems.TheGalerkinmethodandtheRitzmethodofthevariationalproblemswillbedescribedinthelastsection.
The material of Section 1.
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